Ordinal-pattern transition networks as a symbolic substrate for reservoir computing
Abstract
Reservoir computing has established itself as a computationally efficient alternative to fully trained recurrent neural networks using a fixed high-dimensional nonlinear substrate to map temporal input into a feature space for a simple readout. Here, we argue that reservoir computing can be realized entirely within a coarse-grained, symbolic state space, and that constrained random walks on ordinal-pattern transition networks provide a natural instance of this realization, replacing the reservoir's continuous state space with a discrete and directly data-derived one. We develop a formal correspondence between the two frameworks, show that the echo-state property has a natural discrete analog in the ergodicity and spectral gap of the walk's underlying Markov chain, and illustrate the correspondence with a minimal example and an empirical one-step-ahead prediction task evaluated across five paradigmatic model systems. These systems exhibit dynamics that correspond to strictly periodic, quasiperiodic, chaotic, or stochastic signals. We argue that this symbolic reservoir perspective allows a redefinition of the reservoir substrate itself, replacing the requirement of a continuous, randomly generated dynamical system with any sufficiently ergodic, data-derived symbolic process. This opens a route toward interpretable, data-adaptive reservoirs whose internal structure is inferred rather than randomly assigned, with implications for surrogate-data methods and reservoir design more broadly.
Versions
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BibTeX
@misc{potratzki2026ordinalpattern,
title = {{Ordinal-pattern transition networks as a symbolic substrate for reservoir computing}},
author = {Max Potratzki and Thomas G. de Jong and Michael Small and Timo Bröhl},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.21973},
eprint = {2610.21973},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.21973},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Ordinal-pattern transition networks as a symbolic substrate for reservoir computing
AU - Max Potratzki
AU - Thomas G. de Jong
AU - Michael Small
AU - Timo Bröhl
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.21973
UR - https://arxiv.org/abs/2610.21973
AB - Reservoir computing has established itself as a computationally efficient alternative to fully trained recurrent neural networks using a fixed high-dimensional nonlinear substrate to map temporal input into a feature space for a simple readout. Here, we argue that reservoir computing can be realized entirely within a coarse-grained, symbolic state space, and that constrained random walks on ordinal-pattern transition networks provide a natural instance of this realization, replacing the reservoir's continuous state space with a discrete and directly data-derived one. We develop a formal correspondence between the two frameworks, show that the echo-state property has a natural discrete analog in the ergodicity and spectral gap of the walk's underlying Markov chain, and illustrate the correspondence with a minimal example and an empirical one-step-ahead prediction task evaluated across five paradigmatic model systems. These systems exhibit dynamics that correspond to strictly periodic, quasiperiodic, chaotic, or stochastic signals. We argue that this symbolic reservoir perspective allows a redefinition of the reservoir substrate itself, replacing the requirement of a continuous, randomly generated dynamical system with any sufficiently ergodic, data-derived symbolic process. This opens a route toward interpretable, data-adaptive reservoirs whose internal structure is inferred rather than randomly assigned, with implications for surrogate-data methods and reservoir design more broadly.
ER -